Diffusion Models for Cayley Graphs

TL;DR

This paper reframes navigation problems on Cayley graphs, including generalized Rubiks cubes, as diffusion processes with a forward exploration and a backward search toward a goal. A score-based neural model  sigma_{ heta,t} is learned to approximate the score between consecutive states, enabling a reverse diffusion to reach the target from a starting state. The authors introduce a reversed score ansatz that directly ties the forward transition to the learned reverse scores, leading to improved exploration and shorter solution paths in experiments. They demonstrate the approach on Rubiks cubes and a SL_2(z_p) instance, show how varying the forward process can enhance exploration, and discuss extensions to group actions and multiple targets with avenues for future work.

Abstract

We review the problem of finding paths in Cayley graphs of groups and group actions, using the Rubik's cube as an example, and we list several more examples of significant mathematical interest. We then show how to formulate these problems in the framework of diffusion models. The exploration of the graph is carried out by the forward process, while finding the target nodes is done by the inverse backward process. This systematizes the discussion and suggests many generalizations. To improve exploration, we propose a ``reversed score'' ansatz which substantially improves over previous comparable algorithms.

Figures (5)

• Figure 1: Average solution length against number of nodes expanded, for 1000 initial states randomly scrambled between 1000-10000 times. We report results with a ball radius $R=5$, and no $T$-calibration. For beam width $2^7$ or greater, we are able to solve all states.
• Figure 2: Average solution length vs beam width. We find a best fit given by $E = 41.9 log_2(B)^{-1}-1.92$, where $B$ is the beam width (trivially related to the number of nodes). Excesses are always even due to parity. We report results with a ball radius $R=5$, and no $T$-calibration.
• Figure 3: We observe smooth dependence of performance on beam width and radius of the hashed ball in the two-sided search. Of particular note is the significant degradation for zero ball size. Again, we conjecture this is related to the issue of $T$-calibration.
• Figure 4: Even at beam width $18$, with 1 goal state we failed to solve all problems, so we present the mean just for illustration. With 8 goal states, we only failed to solve all states at beam width $2^{14}$ (not plotted).
• Figure 5: Dotted line is the true average distance. Plotted with the radius of the hashed target state ball, $R$. Varying effect of T-calibration, 1000 samples.